The proposed research focuses on two area of statistical methodology related to medical research, particularly cancer research. The first area is that of the design and analysis of Phase I clinical trials. Various designs and analytic approaches will be compared to the traditional dose-escalation scheme and to each other. The goal will be to develop a quantitatively rigorous basis for determining the MTD (Maximum Tolerable Dose) with both a point estimate and confidence interval while, at the same time, maintaining an efficient, conservative and simple design for the trial. Probabilistic properties of the trial will be assessed using a Markov chain representation; other aspects of design and analysis will be assessed largely through Monte Carlo simulation experiments. Another area of research will focus on further development of diagnostic methodologies for the Cox regression model for survival time. The goal will be to develop techniques, analogous to those available for linear regression, that enable one to detect data points that unduly influence the outcome of an analysis or that are not adequately explained by an analysis (have a large residual). Such techniques will improve the ability of investigators to assess the adequacy and robustness of covariate models for survival time, such as those adjusting treatment comparisons for known prognostic factors or seeking to identify new prognostic factors.